Why not have a physical button? Several companies have released programmable buttons. You could also have an emergency-only prepaid cell phone and only give the number to those people. There's also the traditional Life Alert type options that would summon first responders.
physical buttons are hard to debug! emergency-only cellphone looks like the better alternative for now even if it would require me to use two phones or swap sims. It's not a life risk situation, more a (business and peace of mind) life risk situation so i don't think ~life alert could work
Take a look at particle.io. They have a super simple button that you can connect via WiFi or GSM to ifttt or twilio really easily. Sample code leaves almost nothing to debug.